wavelet estimation via sparse multichannel blind deconvolutionkazemino/papers/wavelet... · 2014....

75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London, UK, 10-13 June 2013

Tu P12 12Wavelet Estimation via Sparse Multichannel BlindDeconvolutionN. Kazemi Nojadeh* (University of Alberta) & M.D. Sacchi (university ofAlberta)

SUMMARYWe introduce a Sparse Multichannel Blind Deconvolution (SMBD) method.The method is a modification of Euclid’s blind deconvolution where the multichannelimpulse response is estimated by solving an homogeneous system of equations. SMBD can toleratemoderate levels of noise and does not require knowing the source duration. SMBD solves thehomogeneous system of equations arising in Euclid deconvolution by imposing sparsity on themultichannelimpulse response. To avoid the trivial solution, the sparse reflectivity is constrained to haveunit norm. The latter results in non- convex optimization that is solved via a constrained steepestdescent technique.


Introduction

Deconvolution is an important and recurrent topic in seismic data processing. Many signalsand images can be represented via the convolution of an unknown signal of interest and asource signature. In general, the unknown signal or image can be estimated via inverse meth-ods when the source signature is known. This process is called deconvolution. When thewavelet is unknown the process requires the simultaneous estimation of two signals. This pro-cess is often called blind deconvolution (Shalvi and Weinstein, 1990). Two early attempts onblind deconvolution are Homomorphic Deconvolution based on the work by Oppenheim andSchafer (1968) and implemented for the first time in exploration seismology by Ulrych (1971).Another technique for blind deconvolution was proposed by Wiggins (1985) who coined thename Minimum Entropy Deconvolution (MED) algorithm. MED assumes that the reflectivity issparse and operates by finding an inverse filter that maximizes a measure of sparsity (Donoho,1981). Both Homomorphic Deconvolution and MED suffer from a variety of shortcomings. Forinstance, homomorphic deconvolution is inclined to instability due to phase unwrapping and byits inherent inability to incorporate an additive noise term in the data. MED deconvolution oftentends to annihilate small reflection coefficients (Ooe and Ulrych, 1979; Walden, 1985).

The history of seismic deconvolution is plagued by interesting statistical methods for blinddeconvolution. These methods, however, only work under ideal signal conditions. For instance,important excitement was generated by methods based on higher order statistics because oftheir ability to estimate phase (Petropulu and Nikias, 1992, 1993; Lazear, 1993; Hargreaves,1994; Sacchi et al., 1996; Velis and Ulrych, 1996). However, the conditions for robust waveletestimation required by these methods are not often satisfied by seismic data (Stogioglou et al.,1996).

Euclid deconvolution is a member of the plethora of methods that have been proposed for blinddeconvolution of seismic data. The method was first discussed in the geophysical literature byRietsch (1997a) and tested with real data examples in Rietsch (1997b). The method has alsobeen investigated by Xu et al. (1995) and Liu and Malvar (2001) in communication theory anddeconvolution of reverberations. The idea can be summarized as finding common factors ofthe z-transform of the source embedded in a group of seismograms with different reflectivitysequences. The problem leads to the estimation of the multichannel seismic reflectivity viathe solution of homogeneous system of equations (Mazzucchell and Spagnolini, 2001). In theideal case, the eigenvector associated to the minimum non-zero eigenvalue of the homogenoussystem of equations is an estimator of the multichannel reflectivity. However, small amounts ofnoise impinge on the identification of the eigenvector associated to the impulse response. Wepropose an improvement to Euclid deconvolution where the homogeneous equation is satisfiedby a sparse solution (sparse impulse responses). In other words, we are assuming that themultichannel impulse response of the earth is a multichannel sparse series.

Theory

The earth can be modeled as a linear time invariant (LTI) system. The input-output relationshipfor this system, assuming a stationary source wavelet and a noise free condition, can be writtenas

d j[n] = ∑k

w[n− k]r j[k] , j = 1 . . .J (1)

where the multichannel seismic data can be represented by the d j = (d j[0],d j[2], . . . ,d j[N−1])T ,similarly the impulse responses for channel j can be written as r j = (r j[0],r j[2], . . . ,r j[M −


1])T , and finally we represent the seismic source function (the wavelet) via the vector w =(w[0],w[2], . . . ,w[L− 1])T . We stress that N = M +L− 1. We also remind the readers that con-volution can be expressed using the z-transform as follows

D j(z) =W (z)R j(z) , j = 1 . . .J . (2)

By virtue of equation 2, it is easy to show that

Dp(z)Rq(z)−Dq(z)Rp(z) = 0 , ∀ p,q (3)

which can be rewritten in matrix-vector form as follows

Dp rq −Dq rp = 0 (4)

where Dp and Dq in equation 4 are the convolution matrices of channels p and q, respectively.Combining all possible equations of type 4 leads to the following homogeneous system ofequations

A x = 0 (5)

where

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

D2 −D1D3 −D1D4 −D1...

. . .D3 −D2D4 −D2...

. . .DJ −DJ−2

DJ −DJ−1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)

andx = [r1,r2,r3, . . . ,rJ]T . (7)

Euclid method estimates the reflectivity by estimating the eigenvector associated to the mini-mum non-zero eigenvalue of AT A Rietsch (1997a). A small amount of noise in the data makesthe solution impractical for real data applications Rietsch (1997b). The addition of a noise termin our signal model leads to

Ax = e . (8)

It can be shown that e is white and we propose to find a solution x that minimizes e. In additionwe will constraint x to be sparse. To avoid the trivial solution, we must equip our problem withan extra constraint xT x = 1. We propose to find the solution by minimizing the cost J(x)

x̂ = argminx

J(x), subject to xT x = 1 (9)

whereJ(x) =

12||Ax||22 +λ Hμ(x) . (10)

The symbol Hμ(.) is used to indicate the Huber norm with Huber parameter μ. The Hubernorm forces solutions that are sparse. Using the Huber norm enables us to use a simpleoptimization method based on steepest descent techniques. It is worth mentioning that topreserve the constraint and keep the solution on the unit sphere, one should use an educatedstep that can be derived from Rodrigues’ rotation formula Murray et al. (1994).


ExamplesTo examine the performance of proposed method we applied SMBD to a near offset sectionfrom the Gulf of Mexico seismic data (Figure 1a). Figures 1a and b show the data beforeand after deconvolution via SMBD, respectively. We also estimated the wavelet using thefrequency-domain least squares estimator. The wavelet is portrayed in Figure 2b. For com-pleteness, the average sea floor first break source wavelet is extracted and compared with theestimated wavelet. There is a strong resemblance of the estimated wavelet with the averagefirst break pulse that was extracted by flattening and averaging the water bottom reflection.

Conclusions

We have presented a modification to Euclid’s blind deconvolution method. The proposedmethod alleviates some of the problems encountered in blind deconvolution via Euclid method.For instance, SMBD can tolerate moderate levels of noise and does not require to know theprecise length of the source duration. The method permits to estimate broad-band (sparse)impulse responses that can be used for wavelet estimation.

AcknowledgementsThe authors are grateful to the sponsors of Signal Analysis and Imaging Group (SAIG) at theUniversity of Alberta.

References

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a)

2

4

6

Tim

e (s

)

100 200 300 400 500 600 700 800Trace number

b)

2

4

6

Tim

e (s

)

100 200 300 400 500 600 700 800Trace number

Figure 1 Deconvolution of a Gulf of Mexico near offset section with the estimated wavelet. a)Before deconvolution. b) After deconvolution.

0.05 0.1

−1

−0.5

0

0.5

1

Time (s)

Am

plitu

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Estimated waveletFirst break wavelet

Figure 2 :Comparison of the extracted wavelet from a Gulf of Mexico seismic data using SMBDmethod with the first break pulse.

wavelet estimation via sparse multichannel blind deconvolutionkazemino/papers/wavelet... · 2014....

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